The Dirac-Kohn-Sham scheme

The relativistic generalisation of the Hohenberg-Kohn theorem states that the external four-potential is - except for a gauge transformation - determined by the four-current of the system. The first component e7° is the charge density while the spatial components, J are associated both with orbital currents and the spin density. In the non-relativistic limit, the coupling of electron spin to an external magnetic field is automatically retrieved. 

The Kohn-Sham recipe can be transferred to the relativistic case in complete analogy [11, 12], To arrive at a computational scheme suitable for quantum chemical calculations, it is mandatory to neglect 

It is instructive to look at the non-relativistic limit of j. In the absence of external magnetic fields, the relation between the large and small components, (pj and Xi is, in the non-relativistic limit 

The first term of Eq. (6) is just the non-relativistic current density for a spinless particle, while the second term arises from the electron spin and has the form of the curl of the spin density. In the relativistic case, this decomposition is at best an approximation since spin and linear motion of a particle are coupled. The kinetic energy (including the rest energy) of the Kohn-Sham reference svstem is given 

We use the standard Dirac representation with the 4x4 Dirac matrices 

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The combination of the Dirac-Kohn-Sham scheme with non-relativis-tic exchange-correlation functionals is sometimes termed the Dirac-Slater approach, since the first implementations for atoms [13] and molecules [14] used the Xa exchange functional. Because of the four-component (Dirac) structure, such methods are sometimes called fully relativistic although the electron interaction is treated without any relativistic corrections, and almost no results of relativistic density functional theory in its narrower sense [7] are included. For valence properties at least, the four-component structure of the effective one-particle equations is much more important than relativistic corrections to the functional itself. This is not really a surprise given the success of the Dirac-Coulomb operator in wave function based relativistic ab initio theory. Therefore a major part of the applications of relativistic density functional theory is done performed non-rela-tivistic functionals. 

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